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- Jagers, Peter, 1941, et al.
(author)
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Branching processes with deteriorating random environments
- 2002
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In: Journal of Applied Probability. - 0021-9002 .- 1475-6072. ; 39, s. 395-401
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Journal article (peer-reviewed)abstract
- We introduce Galton-Watson style branching processes in random environments which are deteriorating rather than stationary or independent. Some primary results on process growth and extinction probability are shown. Two simple examples are given.
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- Jagers, Peter, 1941, et al.
(author)
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Convergence to the coalescent in populations of substantially varying size.
- 2004
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In: J. Appl. Probab.. - 0021-9002. ; 41:2, s. 368-378
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Journal article (peer-reviewed)abstract
- Kingman's classical coalescent uncovers the basic pattern of genealogical trees of random samples of individuals in large but time-constant populations. Time is viewed as discrete and identified with non-overlapping generations. Reproduction can be very generally taken as exchangeable (meaning that the labelling of individuals in each generation carries no significance). Recent generalisations have dealt with population sizes exhibiting given deterministic or (minor) random fluctuations. We consider population sizes which constitute a stationary Markov chain, explicitly allowing large fluctuations in short times. Convergence of the genealogical tree, as population size tends to infinity, towards the (time-scaled) coalescent is simply proved under minimal conditions. As a result, a formula for effective population size obtains, generalising the well-knownharmonic mean expression for effective size.
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| 6. |
- Jagers, Peter, 1941, et al.
(author)
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Random variation and concentration effects in PCR
- 2003
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In: J. Theoret. Biol. 224, 299-304 (2003). - 0022-5193 .- 1095-8541. ; 224, s. 299-304
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Journal article (peer-reviewed)abstract
- Even though the efficiency of the PCR reaction decreases, analyses are made in terms of Galton-Watson processes, or simple deterministic models with constant replication probability (efficiency).Recently Schnell and Mendoza have suggested that the form of the efficiency can be derived from enzyme kinetics. This results in the sequence of molecules numbers forming a stochastic process with the properties of a branching process with population size dependence, which is supercritical, but has a mean reproductionnumber that approaches one. Such processes display ultimate linear growth, after an initial exponential phase, as is the case in PCR. It is also shown that the resulting stochastic process for a large Michaelis Menten constant behaves like the deterministic sequence x_n arising by iterations of the function f(x) = x+x/(1+x).
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| 7. |
- Jagers, Peter, 1941, et al.
(author)
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Stochastic fixed points for the maximum
- 2004
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In: Trends in Mathematics} Birkh\"auser Verlag, Basel (2004).
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Journal article (peer-reviewed)abstract
- We consider stochastic fixed point equations X=sup_iT_iX_i (in distribution) in Xfor known $T=(T_1,T_2,... The rvs T,X_i,i = 1, 2, .. are independent and X_i distributed as X. We present a systematic approach in order to find solutions using the monotonicity of the corresponding operator. These equations come up in the natural setting of weighted trees with finite or countable many branches. Examples are in branching processes and the analysis of algorithms (for parallel computing).
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